Arch Calculator

How to Use the Arch Calculator

Using our arch calculator is simple and straightforward. Whether you’re a professional builder, architect, or DIY enthusiast working on an archway project, this tool helps you quickly determine the critical dimensions needed for accurate construction.

Step 1: Choose your preferred unit system (metric or imperial) using the unit switcher buttons. This arch span calculator supports millimeters, centimeters, meters, inches, and feet.

Step 2: Enter the span (total width) of your arch opening. This is the horizontal distance from one side of the arch to the other at the spring line where the arch begins.

Step 3: Input the rise (height) of your arch. This is the vertical distance from the spring line to the highest point of the arch. Remember, for most circular arches, the rise should be less than half the span.

Step 4: Click the “Calculate” button and instantly receive your results, including the radius, focus point distances for elliptical arches, arc length, and more. The live diagram updates automatically to show your arch’s profile.

For advanced users, switch to Advanced Mode to calculate from radius, chord length, or central angle measurements. This arch radius calculator provides flexibility for various architectural and construction scenarios.

What is an Arch? Types & Terminology

An arch is a curved structural element that spans an opening and supports the weight above it. Arches have been used in construction for thousands of years, from ancient Roman aqueducts to modern architectural masterpieces. Understanding arch terminology is essential for accurate calculations and successful building projects.

Common Arch Types

Circular (Semicircular) Arch: The most traditional form, featuring a perfect semicircle. The rise equals half the span, creating a classic rounded appearance commonly seen in Roman architecture.

Elliptical Arch: Uses an ellipse curve rather than a circle, allowing for varying rise-to-span ratios. This arch type is popular in bridges and decorative archways when you need a flatter or taller profile than a semicircular arch provides.

Segmental Arch: A circular arc that’s less than a semicircle, creating a flatter profile. The rise is less than half the span, making it ideal for low-clearance situations.

Gothic (Pointed) Arch: Features two curved segments meeting at a point, distributing weight more efficiently. Common in Gothic cathedral architecture and providing greater height with the same span.

Key Terminology

Span: The horizontal width of the arch opening, measured at the spring line. This is the distance your arch must bridge.

Rise: The vertical height from the spring line to the apex (highest point) of the arch. The rise and span relationship determines the arch’s curvature.

Radius: For circular arches, the distance from the center point to the arch curve. Our elliptical arch calculator can determine this critical measurement.

Spring Line: The horizontal line where the arch begins to curve upward from its supports or abutments.

Focus Points: For elliptical arches, two specific points along the major axis that define the ellipse geometry. These are essential for accurately drawing an arch.

Chord: The straight-line distance across the arch opening, typically at the base.

Sagitta: Another term for the rise, particularly when dealing with circular segments.

Inputs & What They Mean

Understanding what to input into the arch calculator ensures accurate results for your construction or design project.

Span / Width

This represents the total horizontal opening your arch must cover. Measure carefully at the point where the arch will begin (the spring line). For a doorway arch, this would be the door width. For a bridge arch, it’s the distance to be spanned. Accuracy here is critical—even small measurement errors compound when calculating radius and arc length.

Rise / Height

The rise determines how tall your arch will be at its peak above the spring line. This dimension affects both aesthetics and structural properties. A higher rise creates a more dramatic arch but requires more vertical space. For load-bearing arches, the rise influences weight distribution. When using this arch rise calculator, consider your space constraints and desired visual effect.

Unit Selection

Choose units that match your measuring tools and project requirements. Construction professionals often use feet and inches in the United States, while metric measurements (millimeters and centimeters) are standard in most other regions. The calculator automatically converts between systems, but maintaining consistency in your project documentation prevents errors.

Common Mistakes to Avoid

Don’t confuse the rise with the total height of the archway—rise is measured from the spring line, not from ground level. Ensure your rise is realistic for your span; extremely flat arches (very low rise relative to span) can be structurally weak, while very tall rises may not actually form a practical arch. When building an archway, always verify measurements before cutting materials.

Results Interpretation: Rise, Span, Radius, Focus Points

After clicking calculate, you’ll receive several key measurements. Here’s how to understand and use each result from this arch calculator tool.

Calculated Radius

For circular and segmental arches, the radius indicates the distance from the center point to the curve. This is your compass setting if drawing the arch manually. The formula used is: R = (rise² + (span/2)²) / (2 × rise). Larger radii create flatter arches, while smaller radii produce steeper curves. Use this measurement to set up your trammel or large compass when marking the arch curve on materials.

Focus Point Distances

For elliptical arches, you’ll receive two focus points. These are specific locations along the major axis (span) that define the ellipse. To draw an elliptical arch, place pins at these focus points, loop a string around both, and trace with a pencil keeping the string taut—this creates a perfect ellipse. The calculator provides the distance of each focus from the center of the span.

Arc Length

This tells you the actual curved distance along the arch from one spring point to the other. Arc length is essential for estimating materials—if you’re laying brick or stone along the arch curve, you’ll need this measurement to calculate how many units you need. It’s always longer than the span because it follows the curve rather than a straight line.

Chord Length

The straight-line distance across the base of the arch. For most basic applications, this equals your span, but in advanced calculations involving partial arches or specific segments, the chord may differ.

Applying the Results

Use these calculated dimensions to create templates for cutting arch components. For masonry work, the radius helps determine voussoir (arch brick) angles. In carpentry, these measurements guide cutting curved headers and support structures. Always add appropriate allowances for mortar joints, material thickness, and structural requirements specific to your application.

Real-World Examples & Use Cases

This arch calculator serves various practical applications across construction, architecture, carpentry, and DIY projects.

Example 1: Residential Doorway Arch

A homeowner wants to install a decorative arch above a 36-inch (914mm) doorway with a 12-inch (305mm) rise. Using the calculator:

Span: 36 inches
Rise: 12 inches
Result: Radius of approximately 22.5 inches

With this radius, the carpenter can set a trammel or improvised compass to mark the arch curve on the header board, ensuring a perfect curve for cutting.

Example 2: Garden Arbor Construction

A landscaper is building a garden archway with a 6-foot (1829mm) span and wants a 2-foot (610mm) rise for a flatter, more elegant profile. The calculator determines a radius of 61 inches (1549mm) and an arc length of 75 inches (1905mm). This arc length helps estimate the amount of bent wood or metal required for the top frame.

Example 3: Masonry Arch Restoration

A restoration specialist needs to rebuild a damaged portion of a historical arch. By measuring the existing span (2.4 meters) and rise (0.9 meters), the elliptical arch calculator provides focus point distances needed to recreate the original elliptical profile accurately, ensuring the repair matches the historic construction.

Example 4: Automotive Workshop Entry

A business owner needs an arch opening for a 16-foot (4877mm) wide garage entrance with a 3-foot (914mm) rise to accommodate vehicle height. The calculator shows a radius of 117 inches (2972mm), helping the structural engineer design the arch support system.

DIY Applications

Home improvement enthusiasts use this tool for building backyard pizza ovens (calculating the oven mouth arch), constructing decorative garden features, creating custom furniture with arched elements, designing theatrical set pieces, and planning built-in bookshelf archways. The calculator’s accuracy helps avoid costly material waste and ensures professional-looking results.

Professional Applications

Architects and engineers employ arch calculators during preliminary design phases, structural analysis, restoration work, and code compliance verification. Contractors use these calculations for accurate material ordering, bid estimation, template creation, and quality control during construction. The ability to quickly verify arch dimensions improves project efficiency and reduces errors.

Frequently Asked Questions (FAQ)

What is the difference between rise and span in an arch?

The span is the total horizontal width of the arch opening, measured at the base where the arch begins (the spring line). The rise is the vertical height from that spring line to the highest point of the arch (the apex). Think of span as “width” and rise as “height” of the arch curve.

How do I calculate the radius of an arch?

For a circular arch, the radius can be calculated using the formula: R = (rise² + (span/2)²) / (2 × rise). Our calculator does this automatically. Simply input your span and rise measurements, and the radius is computed instantly along with other useful dimensions.

Can I use this calculator for elliptical arches?

Yes! The calculator supports both circular and elliptical arches. For elliptical arches, it provides focus point distances which are essential for accurately drawing the ellipse curve. These focus points allow you to use the “string and pins” method to trace a perfect elliptical arch.

What’s the ideal rise-to-span ratio for an arch?

This depends on your arch type and purpose. Semicircular arches have a 1:2 ratio (rise = half the span). Segmental arches typically use ratios between 1:4 and 1:6 for flatter profiles. Elliptical arches offer flexibility with ratios from 1:3 to 1:8. Structural requirements, aesthetic preferences, and space constraints influence the ideal ratio for your project.

How accurate do my measurements need to be?

For professional construction, aim for accuracy within 1/16 inch (1-2mm). Small measurement errors in span or rise can significantly affect the calculated radius and arc length, potentially causing fit problems during construction. Always measure twice, and use the most precise tools available for your project scale.

Can I use this calculator for structural engineering?

This calculator provides geometric calculations for arch dimensions but does not perform structural analysis. For load-bearing arches, consult a structural engineer to ensure the design meets safety codes and can support required loads. This tool helps with initial design and layout but isn’t a substitute for professional engineering calculations.

What materials work best for constructing arches?

Traditional materials include brick, stone, and concrete for masonry arches. Modern options include engineered wood, steel, laminated veneer lumber (LVL), and fiberglass. Material choice depends on span, load requirements, appearance, and budget. Larger spans or heavy loads typically require stronger materials or reinforcement.

How do I draw an arch from the calculated measurements?

For circular arches, use the calculated radius to set a compass or trammel. Position the center point at the proper location (calculated from the spring line), then trace the arc. For elliptical arches, place pins at the calculated focus points, loop string around both with a pencil, and trace keeping the string taut.

Related Calculators & Resources

Explore our other calculation tools to help with your construction and design projects:

Ellipse Calculator Circular Arc Calculator Angle Calculator Triangle Calculator

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